Abstract : This study examines the simulation of three torrential rain events observed on 13-14 October 1995 (the Cévennes case), 12-13 November 1999 (the Aude case) and 8-9 September 2002 (the Gard case) over the southeastern part of France using the Meso-NH non-hydrostatic mesoscale numerical model. These cases were associated with extreme Heavy Precipitation Events (HPEs) with significant precipitation amounts exceeding 500 mm in less than 24 hours. Several sets of numerical experiments were performed with 10 km and 2.5 km horizontal resolutions. In part I of this study, special attention is paid to the experimental design for obtaining realistic simulations of HPEs with the Meso-NH model, as well as the evolution of the synoptic patterns in which the rainfall events are embedded. The best 2.5 km numerical simulations show the ability of the Meso-NH model to reproduce significant quasi-stationary rainfall events. Moreover, the model fairly reproduces the low-level mesoscale environments associated with the three HPEs. The HPEs formed in a slow-evolving synoptic environment favourable for the development of convective systems (diffluent upper-level southerly flow, PV anomalies, etc.). At lower levels, a southerly to easterly moderate to intense flow provided conditionally unstable and moist air as it moved over the relatively warm Mediterranean Sea, typical for this time of the year (late summer and autumn). The two extreme cases (Gard and Aude) differ from the more classical event (Cévennes) in terms of larger low-level moisture fluxes. Weaker values of conditional convective instability, as in the Aude case, is counterbalanced by a stronger warm and moist low-level jet. The mesoscale triggering and/or sustaining ingredients for deep convection and the physical mechanisms leading to the stationarity of these rainfall events are presented and discussed in a companion paper.